Unique bounded measure realizing the critical polynomial moments

For every prime natural number $p$, every positive natural number $N$, every natural number $k\ge 2$, and every pair of unital ring homomorphisms $\iota:\overline{\mathbb Q}\to\mathbb C$ and $\iota_p:\overline{\mathbb Q}\to\mathbb C_p$, where $\overline{\mathbb Q}$ is the algebraic closure of $\mathbb Q$, suppose the following data are given. There is a holomorphic cusp form $f$ of weight $k$ for $\Gamma_1(N)$ (acting through its image in $\mathrm{GL}_2(\mathbb R)$), a Dirichlet character $\varepsilon$ modulo $N$ with values in $\overline{\mathbb Q}$, and coefficients $c_n\in\overline{\mathbb Q}$ for all natural numbers $n$, such that the coefficient of degree $n$ in the width-one Fourier expansion of $f$ is $\iota(c_n)$, $c_1=1$, and, for every $\gamma=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in\Gamma_0(N)$ and every $z$ in the complex upper half-plane, $f(\gamma z)=\iota(\varepsilon(d))(cz+d)^k f(z)$. For every prime natural number $\ell$ and every such $z$, these data also satisfy $\ell^{-1}\sum_{b=0}^{\ell-1}f((z+b)/\ell)+\iota(\varepsilon(\ell))\ell^{k-1}f(\ell z)=\iota(c_\ell)f(z)$. Further suppose a period system is given: complex numbers $\Omega_u\ne0$ for each Boolean $u$, and algebraic numbers $v_{u,t}(r)$ for every Boolean $u$, natural number $t$, and rational number $r$, such that, whenever $t\le k-2$, $\iota(v_{u,t}(r))=(I_t(r)+\sigma_u(-1)^tI_t(-r))/(2\Omega_u)$, where $\sigma_{\mathrm{true}}=1$, $\sigma_{\mathrm{false}}=-1$, and $I_t(r)=2\pi\int_{(0,\infty)} f(r+iy)(r+iy)^t\,dy$ is the complex Bochner integral against real Lebesgue measure; the $\mathbb Z$-submodule of $\overline{\mathbb Q}$ spanned by all $v_{u,t}(r)$ with $t\le k-2$ is required to be finitely generated. The values with $t>k-2$ are part of the data but have no comparison or lattice condition. Suppose also that $\alpha\in\mathbb C_p$ satisfies $\lVert\alpha\rVert=1$ and $\alpha^2-\iota_p(c_p)\alpha+\iota_p(\varepsilon(p))p^{k-1}=0$. Then, for each Boolean $s$, there exists exactly one continuous $\mathbb C_p$-linear functional $\mu:C(\mathbb Z_p^\times,\mathbb C_p)\to\mathbb C_p$, on the space of continuous functions with its usual uniform topology, with the following property: for every natural number $n>0$, every integer $a$ coprime to $p$, and every natural number $j\le k-2$, there exists a continuous function $g:\mathbb Z_p^\times\to\mathbb C_p$ which is $x^j$ when the underlying $p$-adic integer $x$ reduces to $a$ modulo $p^n$, and is zero otherwise, and this function satisfies $\mu(g)=\alpha^{-n}\iota_p(A_{s,j}(a,p^n))-\bigl(\iota_p(\varepsilon(p))p^{k-2}/\alpha^{n+1}\bigr)\iota_p(A_{s,j}(a,p^{n-1}))$, where, for the nonzero rational numbers $m$ occurring here, $A_{s,j}(a,m)=\sum_{t=0}^j\binom jt m^t a^{j-t}v_{s,t}(-a/m)\in\overline{\mathbb Q}$, and $x$ in $x^j$ denotes the image of the underlying $p$-adic integer under $\mathbb Z_p\hookrightarrow\mathbb Q_p\hookrightarrow\mathbb C_p$. Uniqueness is among all such continuous linear functionals satisfying every displayed moment condition, and asserts equality on every continuous function. The quantifiers include $p=2$, $N=1$, primes dividing $N$, both signs independently, all positive and negative integer representatives $a$ coprime to $p$, $n=1$ (where the second symbol has denominator $p^0=1$), and $j=0$; for $k=2$ only $j=0$ is prescribed and $p^{k-2}=1$. No moment at depth zero is prescribed. The natural-number subtractions in the formulas are ordinary subtractions in the stated ranges, and all displayed divisors $m$, $\Omega_u$, and $\alpha$ are nonzero under the hypotheses. Bochner integrals use the total integral convention, assigning zero if integrability fails; no separate integrability hypothesis is included. The conclusion is conditional on the existence of all the stated form, period, and root data and does not assert their existence.